Subgroups — #20295774

Order 6976.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16812687	C₂	1	2	yes	maximal, complemented
2	#16812684	C₂	1	4	yes	maximal, complemented
3	#16812677	C₂	1	4	yes	maximal, complemented
4	#16812674	C₂	1	4	yes	maximal, complemented
5	#125×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₂₁₈	C₂²	1	2	yes	—
7	C₂×D₄₃₆	C₂²	1	4	yes	—
8	C₂×D₄₃₆	C₂²	1	2	yes	—
9	C₂×C₄×D₁₀₉	C₂²	1	4	yes	—
10	#13182008	C₂²	1	4	yes	—
11	#13182006	C₂²	1	4	yes	—
12	#13182006	C₂²	1	4	yes	—
13	#13182006	C₂²	1	4	yes	—
14	C₄⋊C₄×C₁₀₉	C₂²	1	4	yes	—
15	C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
16	C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂²×D₁₀₉	C₂³	1	4	yes	—
18	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
19	C₂×C₄×C₁₀₉	C₂³	1	2	yes	—
20	C₂×C₄×C₁₀₉	C₂³	1	4	yes	—
21	C₂×Dic₁₀₉	C₂³	1	4	yes	—
22	D₄₃₆	—	4	2	no	—
23	D₄₃₆	—	4	4	no	—
24	C₄×D₁₀₉	—	4	4	no	—
25	Dic₂₁₈	—	4	2	no	—
26	C₂²×C₁₀₉	C₂⁴	1	1	yes	characteristic
27	C₂×D₁₀₉	—	2	4	no	—
28	C₂×D₁₀₉	—	2	4	no	—
29	C₂×D₁₀₉	—	2	4	no	—
30	C₄×C₁₀₉	—	2	2	no	—
31	C₄×C₁₀₉	—	2	4	no	—
32	C₄×C₁₀₉	—	2	1	no	—
33	Dic₁₀₉	—	2	4	no	—
34	C₂×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
35	C₂×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
36	D₁₀₉	—	4	4	no	—
37	C₁₀₉	#562	1	1	yes	characteristic, complemented, Sylow
38	#562	—	109	1	no	maximal, Sylow
39	#125	D₁₀₉	1	1	yes	characteristic, complemented
40	#124	—	109	2	no	—
41	#123	—	109	4	no	—
42	C₂²⋊Q₈	—	109	4	no	—
43	#121	—	109	4	no	—
44	C₂×Q₈	—	109	2	no	—
45	C₂×D₄	—	109	4	no	—
46	C₂×D₄	—	109	2	no	—
47	C₂²×C₄	—	109	4	no	—
48	C₄⋊C₄	—	109	4	no	—
49	C₄⋊C₄	C₂×D₁₀₉	1	4	yes	—
50	C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
51	C₂²⋊C₄	—	109	4	no	—
52	C₂²⋊C₄	—	109	4	no	—
53	C₂²⋊C₄	—	109	4	no	—
54	C₄²	C₂×D₁₀₉	1	1	yes	characteristic
55	C₂³	—	109	4	no	—
56	Q₈	—	436	2	no	—
57	D₄	—	436	4	no	—
58	D₄	—	436	2	no	—
59	C₂×C₄	—	109	4	no	—
60	C₂×C₄	—	436	4	no	—
61	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₁₀₉	1	2	yes	—
63	C₂×C₄	C₂²×D₁₀₉	1	4	yes	—
64	C₂²	—	218	4	no	—
65	C₂²	—	218	4	no	—
66	C₂²	—	218	4	no	—
67	C₂²	C₂³×D₁₀₉	1	1	yes	characteristic
68	C₄	—	2	4	no	—
69	C₄	—	2	2	no	—
70	C₄	—	2	1	no	—
71	C₄	—	218	4	no	—
72	C₂	—	436	4	no	—
73	C₂	#16812745	1	2	yes	—
74	C₂	#16812747	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂⁴
Fitting	#125×C₁₀₉	C₂
hypercenter	#125	D₁₀₉
hypocenter	C₁₀₉	#562
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ #125

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #16812687	C₂	semidirect
2. #16812684	C₂	semidirect
3. #16812677	C₂	semidirect
4. #16812674	C₂	semidirect
5. #125×C₁₀₉	C₂	semidirect
37. C₁₀₉	#562	semidirect
39. #125	D₁₀₉	semidirect